IB Math HL – Lesson 14DEF
Name _______________________________ Block: ______
Vector Magnitude, Operations with Vectors, and the Vector Between Two Points
Algebraic Length of a Vector: We use the Pythagorean Theorem to determine the length (magnitude) of a vector.
−4
Given vector v = ( ), find |v|.
3
 v1 
 then | v |
 v2 
If v  
 3 
 and r  –i – 2j
  5
Ex1) Given p  
find:
a) | p |
b) | r |
A unit vector is any vector which has a length of one unit.
Ex2) Which of the following are unit vectors?

a) 


1
2
1
2
  13 
b)  2 
 3 




k 
Ex3) Find k given that  2  is a unit vector.
5
  12
13 
c)  5 
  13 
Vector Addition: Draw arrow diagrams to represent a + b. Then represent a, b, and a + b each in component form.
a
a=
b
b=
a+b=
 a1 
b 
 and b   1 
 a2 
 b2 
If a  
then a  b 
Negative Vectors: Draw an arrow diagram to represent -a. Then represent it algebraically (component form).
 3 

  4
a  
-a =
 a1 
 then  a 
 a2 
If a  
Vector Subtraction:
If
Scalar Multiplication:
 v1 

 v2 
If k is a scalar and v  
and
then k v 
then
  2
 3 
1
 , q    , and r    find geometrically:
  6
  2
 4
Ex4) Given p  
a) q – p
b) p – q – r
c)
1
2
r + 2q
 3 
  2
1
 , q    , and r    find algebraically and compare to Ex4.
  2
  6
 4
Ex5) Given p  
a) q – p
b) p – q – r
c)
1
2
r + 2q
2
−3
Ex6) Given ⃗⃗⃗⃗⃗
BA = ( ) and ⃗⃗⃗⃗⃗
BC = ( ), find ⃗⃗⃗⃗⃗
AC .
−3
1
 x
A point in the 2-dimensional plane is represented by P(x, y) where OP    is its position vector.
 y
Ex7) Given the points A(2, 4) and B(5, 1)
a) Graph the position vectors a = OA and b = OB .
b) Determine an expression for AB using a and b.
c) Use the information above to determine a formula for the position vector of B relative to A.
Given A(a1, a2) and B(b1, b2),
AB 
Ex8) If A is (4, -3) and B is (-1, 0), find:
a) OA
b) AB
c) BA
Ex9) ABCD is a parallelogram. A is (–1, 2), B is (2, 0) and D is (3, 1). Find the coordinates of C.
Classroom Practice 14DEF:
Exercise 14D: #1d, 2d, 3de, 5
Exercise 14E: #1cgh, 2ac, 4ch, 6e, 7acd, 9
Exercise 14F: #1e, 2a, 4, 5, 7c